On spherical barycentric coordinates

TL;DR

This paper introduces a new method for constructing generalized barycentric coordinates on a sphere, based on 3D barycentric coordinates, revealing properties like edge linearity and Lagrange property, and comparing different coordinate types.

Contribution

It presents a novel construction of spherical barycentric coordinates derived from 3D coordinates, differing from classical planar-based methods, and analyzes their properties and relationships.

Findings

Spherical mean value coordinates from both approaches coincide.

Spherical Wachspress coordinates generally differ between approaches.

Coordinates exhibit properties like edge linearity and Lagrange property.

Abstract

This paper describes a novel construction of generalized barycentric coordinates of points on a sphere with respect to the vertices of a given spherical polygon that is contained in a common hemisphere. While in the standard approach such coordinates are derived from their classical planar counterparts (e.g. Wachspress, or mean value), we instead derive them from 3D barycentric coordinates of the origin and show that they are endowed with some useful properties such as edge linearity and Lagrange property. In addition, we show that spherical mean value coordinates of both approaches coincide while their corresponding spherical wachspress coordinates are in general different.